Coordinate-free formationstabilizationbasedonrelativepositionmeasurements ⋆

Miguel Aranda a, Gonzalo Lopez-Nicolas a, Carlos Sagues a, Michael M. Zavlanos b

aInst. de Investigacion en Ingenierıa de Aragon - Universidad de Zaragoza, C/ Marıa de Luna 1, E-50018 Zaragoza, Spain

bDept. of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, United States

Abstract

This paper presents a method to stabilize a group of agents moving in a two-dimensional space to a desired rigid geometricconfiguration. A common approach is to use information of relative interagent position vectors to carry out this specific controltask. However, existing works in this vein either require the agents to express their measurements in a global coordinatereference, or generally fail to provide global stability guarantees. Our contribution is a globally convergent method that usesrelative position information expressed in each agent’s local reference frame, and can be implemented in a distributed networkedfashion. The proposed control strategy, which is shown to have exponential convergence properties, makes each agent move soas to minimize a cost function that encompasses all the agents in the team and captures the collective control objective. Thecoordinate-free nature of the method emerges through the introduction of a rotation matrix, computed by each agent, in thecost function. We consider that the agents form a nearest-neighbor communications network, and they obtain the requiredrelative position information via multi-hop propagation, which is inherently affected by time-delays. We support the feasibilityof such distributed networked implementation by obtaining global stability guarantees for the formation controller when thesetime-delays are incorporated in the analysis. The performance of our approach is illustrated with simulations.

Key words: Multiagent systems; Formation stabilization; Autonomous mobile robots; Networked control systems;Time-delay systems

1 Introduction

We address in this paper the problem of controlling amultiagent group, which has many interesting applica-tions, e.g. autonomous multivehicle control, cooperativesensing, surveillance, or search and rescue missions. Inparticular, we are interested in the stabilization of a teamof mobile agents to a desired geometric configuration.Diverse methods have been proposed to carry out thistask. Approaches typically rely on defining the multia-gent formation in terms of absolute positions the agentsmust reach (Ren & Atkins, 2007; Zavlanos & Pappas,2007; Dong & Farrell, 2008; Sabattini, Secchi, & Fan-

⋆ This work was supported by Ministerio de Economıa yCompetitividad/European Union (project DPI2012-32100),Ministerio de Educacion (FPU grant AP2009-3430), andNSF (CNS # 1261828 and CNS # 1302284). Correspondingauthor M. Aranda. Tel. +34 876555075.

Email addresses: marandac@unizar.es (Miguel Aranda),gonlopez@unizar.es (Gonzalo Lopez-Nicolas),csagues@unizar.es (Carlos Sagues),michael.zavlanos@duke.edu (Michael M. Zavlanos).

tuzzi, 2011) or in terms of relative quantities such asposition vectors or distances between the agents (Mes-bahi & Egerstedt, 2010). In particular, relative position-based formation stabilization (Lin, Francis, & Maggiore,2005; Olfati-Saber, Fax, & Murray, 2007; Ji & Egerst-edt, 2007; Dimarogonas & Kyriakopoulos, 2008; Cortes,2009; Kan, Dani, Shea, & Dixon, 2012; Coogan & Ar-cak, 2012; Oh & Ahn, 2014) specifies the formation interms of relative distances and bearings between agents.This leads to a unique desired shape and, by using lin-ear consensus-based control laws, permits global stabi-lization when the formation graph (i.e. the graph encap-sulating the interactions between agents) is connected.However, methods based on position information (eitherabsolute or relative) require the agents’ measurementsused for the control to be expressed in a global refer-ence frame. In particular, in relative position-based ap-proaches, the agentsmust share a common sense of orien-tation. For flexibility (i.e. in GPS-denied environments),simplicity and autonomy of the agents, a scenario wherethey can rely on their independent onboard sensors, i.e.use only locally referred information, is interesting.

Preprint submitted to Automatica 16 March 2015

Distance-based formation control (Olfati-Saber & Mur-ray, 2002; Hendrickx, Anderson, Delvenne, & Blondel,2007; Krick, Broucke, & Francis, 2008; Dimarogonas &Johansson, 2009; Oh & Ahn, 2011) addresses this sce-nario. Since the relative bearings between the agents can-not be expressed in a common reference, this methodol-ogy resorts to specifying the formation in terms of inter-agent distances only, and requires the formation graphto be rigid (Anderson, 2008). Still, when generic num-bers of agents are considered, no leader agents are used,and the desired task is for the team to acquire a rigidshape, these schemes provide only local stability guar-antees, even with a complete formation graph. Globalstabilization to a rigid formation is not achievable us-ing negative-gradient, distance-based formation control(Dimarogonas & Johansson, 2009; Anderson, 2011). Inaddition, using distances to specify the formation im-plies that the target shape is always defined up to a re-flection of the pattern, i.e. it is not unique. Even if onlydistances are used in the specification, knowledge of thedirections to the neighboring agents is required in thesemethods to compute the control inputs.

We present here an approach that requires the sameknowledge as distance-based controllers (i.e. locally ex-pressed relative positions), but achieves global stabil-ity. The final positions of the agents form a specifiedrigid shape, defined up to translation and rotation. Thecontrol task is accomplished via minimization of a costfunction defined for each agent in terms of global in-formation, i.e. the relative positions of all other agents.We address the misalignment between orientation refer-ences by introducing in the cost function a local rotationmatrix acting on the relative interagent vectors. Theserotations, on which the agents implicitly agree, capturethe method’s independence of any global reference.

Our multiagent team constitutes a networked system, inwhich the agents interact via communications. If theyhave up-to-date relative position information, we showthat our controller has exponential convergence proper-ties. Realistically, the mobile agents are subject to strictlimitations in terms of power consumption, computa-tional resources and communication ranges, which re-quire the use of a distributed, nearest-neighbor networkscheme. With such setups, system scalability is greatlyenhanced, but multi-hop communication causes the in-formation used by the agents to be affected by signifi-cant time-delays. This effect needs to be introduced inthe model of our system, which then becomes a nonlin-ear time-delay system (Gu, Kharitonov, & Chen, 2003;Richard, 2003) that is, in addition, interconnected (Hua& Guan, 2008; Papachristodoulou, Jadbabaie, & Munz,2010; Nedic & Ozdaglar, 2010). As illustrated by thisrelevant literature, obtaining stability results for sucha system in general conditions (e.g. varying delays, orswitching network topologies) is a complex problem. Wepresent a Lyapunov-based study of our system’s con-vergence, considering that the network’s communica-

tion pattern, in terms of active links and time-delay val-ues, is fixed. Constant point-to-point delays are a usualassumption in the study of nonlinear time-delay sys-tems (Richard, 2003; Papachristodoulou, Jadbabaie, &Munz, 2010), while the consideration of a fixed interac-tion/communication graph topology is ordinary in mul-tiagent formation stabilization methods (Dimarogonas& Kyriakopoulos, 2008; Cortes, 2009; Dimarogonas &Johansson, 2009; Guo, Lin, Cao, & Yan, 2010; Oh &Ahn, 2011). Then, by assuming the agents’ motions sat-isfy certain bounds regarding maximum accelerationsand minimum interagent separations, we establish anupper bound for the system’s worst-case point-to-pointtime-delay such that global stability is ensured.

Other relevant work addressing coordinate-free multia-gent motion using not merely distances includes Lopez-Nicolas, Aranda, Mezouar, and Sagues (2012), where aformation is stabilized using a central coordinator whichemploys visual sensing. In Zhang (2010), each agentcomputes its motion using global information and bothrelative position and velocity information are employed,while here we use only positions. Distributed schemeshave been presented addressing behaviors differentfrom rigid-shape stabilization, e.g. rendezvous (Cortes,Martınez, & Bullo, 2006; Yu, LaValle, & Liberzon,2012), flocking (Jadbabaie, Lin, & Morse, 2003) andother coordinated motion patterns (Moshtagh, Michael,Jadbabaie, & Daniilidis, 2009), or assuming the agentsagree on (and then maintain) a common orientation ref-erence before or during the control execution (Cortes,2009; Oh & Ahn, 2014). Although formation schemesbased on leader agents (Desai, Ostrowski, & Kumar,2001; Guo, Lin, Cao, & Yan, 2010) have also beenvery popular, leaderless approaches, such as the one wepropose, provide greater robustness and flexibility.

To summarize, our contribution is an approach imple-mentable in a distributed networked fashion that glob-ally stabilizes a multiagent group to an arbitrary uniquerigid shape in the absence of a global coordinate system,and not relying on central coordinators or leader agents.

The rest of this paper is organized as follows: In Sec-tion 2 we define the formation control problem and dis-cuss the communication patterns in the underlying net-worked system. In Section 3, we describe the proposedcoordinate-free control method. Section 4 presents thestability analysis of our approach, addressing both a sce-nario where the agents have instantaneous global infor-mation, and the case where they form a distributed net-worked system in which information propagation is af-fected by time-delays. Simulation results are presentedin Section 5. Finally, a brief discussion and the conclu-sion of the paper are given in Section 6.

2

2 Problem formulation

Consider a group of N agents in R2 having single inte-

grator kinematics, i.e. satisfying:

qi = ui, (1)

where qi ∈ R2 denotes the position vector of agent i and

ui ∈ R2 is its control input. We define a desired config-

uration, or formation shape, by a certain, fixed, refer-ence layout of the N agents in their configuration space.The way in which we encode the desired configuration isthrough a set of interagent relative position vectors. Tocapture the existence or absence of an interaction in ourcontrol method between every pair of agents, we definean undirected formation graph, Gf = (V , Ef ), where Vis a set of N vertices, each one associated with an agent,and Ef is a set of links, each one expressing the connec-tion between a pair of agents. Then, for every neighborj of agent i in Gf , we denote as cji ∈ R

2 the vectorfrom i to j in the reference layout of the agents thatdefines the desired configuration. The agents are not in-terchangeable, i.e. each of them has a fixed place in thetarget formation. We then consider that the agents arein the desired configuration if the reference layout hasbeen achieved, up to a rotation and translation.

The problem that we set out to solve in this paper isspecified as follows:

Problem 1 Given an initial configuration in which theagents are in arbitrary positions, the control objective isto stabilize them in a set of final positions such that thegroup is in the desired configuration.

We also define a graph Gc = (V , Ec) to capture the com-munications in our distributed networked system. Theedges Ec express the presence or absence of a direct com-munication link between every pair of nodes (associatedwith agents) in V . Each agent in the system is assumedto be able to obtain an estimation of the relative posi-tions of its set of neighbors in Gf . This is the only infor-mation used by the proposed control strategy, which isdescribed in the following section.

3 Coordinate-free control strategy

We pose the formation control problem as the minimiza-tion of the following cost function for each agent i:

γi =1

4

∑

j∈Ni

∑

k∈Ni

||qjk −Ricjk||2, (2)

where the set Ni includes the neighboring agents of iin the formation graph and agent i as well. In addition,qjk = qj −qk, with the position vectors expressed in an

Fig. 1. Illustration of the cost function γi for agent i. Threeagents i, j and k are considered. γi is defined by the sum ofsquared norms of the differences between the current relativevectors (i.e. qji,qjk,qki in the figure) and the vectors thatencode the desired pattern, (i.e. cji, cjk, cki in the figure),rotated by the matrix Ri. To minimize γi, agent i moves inthe direction of the vector shown with a dashed line. Thecontrol strategy for agents j and k is analogous.

arbitrary global coordinate frame, and Ri ∈ SO(2) is arotation matrix:

Ri =

[

cosαi − sinαi

sinαi cosαi

]

, (3)

where the value of αi is discussed below. The cost func-tion is a sum of squared distances that expresses howseparated the set of agents is from the desired configu-ration. It can be observed that γi accounts for agent i’sdistance to its neighbors, but also for the distances be-tween its neighbors. Also, the local rotation matrix Ri

in (2) is the key component of our method that allows itto be coordinate-free. The geometric meaning of the costfunction is illustrated in Fig. 1. We compute the matrixRi so as to minimize γi. For this, we express the func-tion in terms of αi and the components of the vectors,qjk = [qxjk, q

yjk]

T , cjk = [cxjk, cyjk]

T :

γi =1

4

∑

j∈Ni

∑

k∈Ni

[

(qxjk − cxjk cosαi + cyjk sinαi)2 (4)

+(qyjk − cxjk sinαi − cyjk cosαi)2]

.

To minimize γi with respect to αi, we solve ∂γi

∂αi= 0.

After manipulation, this leads to:

∂γi∂αi

=1

2

[

sinαi

∑

j∈Ni

∑

k∈Ni

(qxjkcxjk + qyjkc

yjk)

− cosαi

∑

j∈Ni

∑

k∈Ni

(−qxjkcyjk + qyjkc

xjk)]

= 0. (5)

3

Solving (5) with respect to the rotation angle αi we get:

αi = arctan

∑

j∈Ni

∑

k∈NiqTjkc

⊥jk

∑

j∈Ni

∑

k∈NiqTjkcjk

. (6)

where c⊥jk = [(0, 1)T , (−1, 0)T ]cjk. Let us define the vari-able Ti ∈ R which will be useful for the analysis of thecontroller throughout the paper:

Ti = tanαi =

∑

j∈Ni

∑

k∈NiqTjkc

⊥jk

∑

j∈Ni

∑

k∈NiqTjkcjk

. (7)

Observe from (6) that there are two possible solutions forαi, separated by π radians. In order to select the correctone, we compute the second order derivative from (5):

∂2γi∂α2

i

=1

2

[

cosαi

∑

j∈Ni

∑

k∈Ni

(qxjkcxjk + qyjkc

yjk)

+ sinαi

∑

j∈Ni

∑

k∈Ni

(−qxjkcyjk + qyjkc

xjk)]

. (8)

By considering together (5) and (8), it can be read-ily seen that one of the solutions of (6) minimizes γi,while the other maximizes the function. The solutionthat is a minimum satisfies the condition ∂2γi

∂α2

i

> 0.

If we isolate the term sinαi in (5) and then substi-tute it in (8), we get that this condition holds whensin(αi)/

∑

j∈Ni

∑

k∈NiqTjkc

⊥jk > 0. From this, it is

straightforward to deduce that the value of αi that min-imizes γi, i.e. the value used in our controller, is givenby:

αi = atan2(∑

j∈Ni

∑

k∈Ni

qTjkc

⊥jk,∑

j∈Ni

∑

k∈Ni

qTjkcjk), (9)

where the atan2 function gives the solution of (6) forwhich αi is in the quadrant that corresponds to thesigns of the two input arguments. Note that the caseatan2(0, 0), for which αi is not defined, is theoreticallypossible for degenerate configurations of the agentswhere γi is constant for all αi, see (5). These degen-eracies are linked to the desired geometry (i.e. are notrelated with our control strategy). They are measurezero, i.e. they will never occur in practice and, therefore,we do not consider them in our analysis.

The control law for agent i is then obtained as the neg-ative gradient of the cost function with respect to qi. Inparticular, by separating the terms in γi depending di-rectly on qi from the part of the function depending onRi, we can express the controller as follows:

qi = Kc

[

−∂γi∂qi

− ∂γi∂αi

∂αi

∂qi

]

= Kc

∑

j∈Ni

qji −Ri

∑

j∈Ni

cji

,

(10)

Fig. 2. Representation of the quantities used in the controllaw for agent i, expressed in an arbitrary global frame G.Three agents i, j and k are depicted, and the local frame Lof agent i is shown.

where we have used that ∂γi

∂αi= 0, and Kc is a positive

control gain.

Let us show that each agent can compute its controlinput in the absence of any global coordinate reference.For this, we denote as θi the rotation angle betweenthe arbitrary global frame and the local frame in whichi operates, and by Pi(θi) ∈ SO(2) the correspondingrotation matrix. Let us now write down the local controllaw computed by i, using a superscript L to denote thatthe quantities are expressed in the local frame:

qLi = Kc

∑

j∈Ni

qLji −RL

i

∑

j∈Ni

cLji

. (11)

Observe that qLi = Piqi. Each agent minimizes γL

i , i.e.the cost function expressed in its local frame. Given thatqLjk = Piqjk for all j, k, we can write, through simple

manipulation:

γLi =

1

4

∑

j∈Ni

∑

k∈Ni

||qLjk −RL

i cjk||2 =

1

4

∑

j∈Ni

∑

k∈Ni

||qjk −P−1i RL

i cjk||2. (12)

Since Ri minimizes γi and RLi minimizes γL

i , and theminimum is unique (as shown earlier in the section),we clearly have from (2) and (12) that γi = γL

i , andthus Ri = P−1

i RLi , i.e. R

Li = PiRi. Therefore, we can

readily see that (11) has the form (10) when expressedin the global frame. Figure 2 provides illustration of thevariables used by the controller with respect to the globaland local frames.

4

4 Stability analysis

We note the similarity of the expression (10) toconsensus-based formation control laws in the litera-ture (Olfati-Saber, Fax, & Murray, 2007; Dimarogonas& Kyriakopoulos, 2008; Mesbahi & Egerstedt, 2010).The difference is given by the rotation matrix that weintroduce. In a scenario where the use of informationis distributed (i.e. Gf is not complete), the proposedcontrol method requires the agents to reach consen-sus in this rotation matrix for the system to be stable.Whether this consensus will occur or not depends onthe formation graph and the geometry of the desiredformation and is not trivial to determine. In this paper,we focus our attention on the case where Gf is complete,i.e. Ni = Nt ∀i, with Nt being the set that containsall the agents. We first assume that up-to-date globalinformation is always available to the agents, i.e., thecommunication graph Gc is also complete. Then, weconsider a distributed scenario where the agents receivethe information in Gf through nearest neighbor commu-nications that are subject to time-delays. In this lattercase, Gc can be any connected graph and the intro-duction of time-delays captures the effect of multi-hopinformation propagation in the networked system.

4.1 Stability with global information

Let us analyze the proposed control method consideringthat all the agents have instantaneous access to com-plete information of the system. We obtain the followingresult:

Theorem 2 Assume that both Gf and Gc are completegraphs. Then the multiagent system evolving according to(10) converges exponentially to the desired configuration.

PROOF. Notice first from equation (7) that when allthe N agents have global information, the value of Ti

will be the same for all of them at any given time in-stant. Let us denote by T (t) this common variable, i.e.T (t) = Ti(t) ∀i and by T0 its initial value. Accordingly,we denote as R(α, t) the common rotation matrix thatthe agents compute, and as R0(α0) the initial one, withα and α0 being the respective rotation angles.

We will compute the time derivative of T , which can beobtained as:

T = Ti =∑

j∈Ni

(

∂Ti

∂qj

)T

qj. (13)

Unless otherwise stated, all the sums that follow in theproof are carried out over the elements ofNt. We use thenomenclature P =

∑

i

∑

j qTij cij, P⊥ =

∑

i

∑

j qTij c

⊥ij .

Thus, T = P⊥/P . From (7), we have the following ex-pression:

T =∑

j

[

P∑

k c⊥jk

T − P⊥

∑

k cjkT

P 2Kc

∑

k

(qkj −Rckj)

]

.

(14)Now, the following identity can be readily shown to betrue by expressing all the vectors with subindex jk interms of their two separate components j, k and thenregrouping them:

∑

j

[

∑

k

cjkT∑

k

qjk

]

=N

2

∑

j

∑

k

qjkT cjk. (15)

By applying the above identity on all four addends inthe numerator of equation (14), we get:

T = Kc

P∑

j

∑

k c⊥jk

T(Rcjk)− P⊥

∑

j

∑

k cjkT (Rcjk)

2P 2/N.

(16)

Given that cjkT (Rcjk) = cos(α)||cjk||2, c⊥jk

T(Rcjk) =

sin(α)||cjk||2, and T = tanα = P⊥/P , it is straightfor-

ward to see that T = 0. Furthermore, we observe thatthe angle α, obtained as atan2(P⊥, P ) (9), must alwaysstay in the same quadrant and is, therefore, constant. Achange of quadrant would imply a jump in the value ofthe angle, which is not possible given that the evolutionof the system is continuous. Therefore, α is constant asthe system evolves, and the rotation matrix computedby the agents remains constant for all time, i.e.R = R0.

Let us now write down the dynamics of the relative po-sition vector between any pair of agents i and j:

qij = qi − qj = Kc

[

∑

k

(qki − qkj)−R0

∑

k

(cki − ckj)

]

=

−KcN · (qij −R0cij). (17)

Thus, we conclude that the multiagent system convergesexponentially to the desired configuration. 2

Remark 3 The proposed controller allows to predict col-lisions. To illustrate this, observe that the predicted evo-lution of the interagent vector qij at a given initial instantt0 has, from (17), the following form:

qij(t) = qij(t0)e−KcN(t−t0) +R0cij

[

1− e−KcN(t−t0)]

.

(18)Thus, qij will e.g. become null at time t = t0 +ln(2)/(KcN) if it is satisfied that qij(t0) and R0cij areparallel, have equal length and lie on opposite sides ofthe coordinate origin. A general collision risk occurswhen the following conditions hold: qij

TR0c⊥ij = 0 and

5

qijTR0cij < 0. Every agent can evaluate these two con-

ditions for all other agents already at the beginning of thecontrol execution. This interesting prediction propertycan facilitate the actual avoidance of collisions. Althoughwe do not address here formal guarantees in this respect,a possible strategy that may be explored is for the agentsthat predict a future collision to modify their controlgains temporarily. The idea is that, then, the resultinggain imbalance among the agents would slightly modifythe rotation matrix computed by the group, changing theagents’ trajectories and preventing the collision.

4.2 Stability in a networked distributed scenario

Next, we analyze the behavior of the networked dis-tributed implementation of the system we propose. Theinformation used by the agents is subject to time-delaysdue to its propagation across the multiagent group, mod-eled by the communications graph Gc. Clearly, the max-imum number of communication hops needed for anagent to get the relative position information of anotheragent is N − 1. Assuming that the delay associated witheach of these hops is bounded by a certain time lapse∆t, the worst-case delay is given by:

D = (N − 1) ·∆t. (19)

The effect of time-delays on the stability of nonlinearcontrol systems, such as the one we propose, is wellknown to be complex (Richard, 2003) and it is oftendifficult to find bounds for the maximum worst-casetime-delay that makes the system stable. In our case,the interconnected nonlinear dynamics and the pres-ence of multiple different delays complicate things fur-ther (Gu, Kharitonov, & Chen, 2003).We present next aLyapunov-based stability study of our control method.

Let us define di =∑

j∈Ntqij −Rcij, where R = Ri ∀i

can be parameterized by T = Ti ∀i, computed as in(7). We denote as τji < D the delay in the transmissionof information from agent j to agent i, not necessarilya direct neighbor of j, across the network. We modelthe effect of this delay in our system as an error in theposition vector measured from i to j, which we call ǫji,expressed as follows:

ǫji(t) = qji(t− τji)− qji(t), ∀i, j ∈ Nt. (20)

We also define ei as the error vector, caused by delays,affecting each agent’s control input (10), such that theactual motion vectors can be expressed as follows:

qi = −Kc · (di − ei) , (21)

and we denote as ||e|| =∑i∈Nt||ei|| the aggregate error

for the complete system. Let us formulate the followingassumptions regarding the characteristics of the system,which will be used in our analysis:

A1: Gc is a fixed connected graph and τji is constant∀t ∀i, j. In other words, the network’s communicationpattern, defined by the graph’s nodes, edges and theirassociated time-delays, is fixed over time. Notice thatthis implies that the evolution of the state of the systemis continuous. In general, the delays between differentagents are not equal, i.e. τji 6= τkl ∀i, j, k, l ∈ Nt if j 6= kor i 6= l.

A2: The magnitude of the total disturbance created bythe time-delays, which is expressed by ||e||, is boundedby the system’s state, i.e. ||e|| < B ·

∑

i ||di|| for someconstant B > 1.

A3: For any given time t, if∑

j∈Nt

∑

k∈Ntqjk

T cjk⊥ = 0

in a given reference frame at t, then there exists avalue M such that |∑j∈Nt

∑

k∈Ntqjk

T cjk| > M inthe same frame. This assumption means that initiallynot all the agents are close together and that theywill remain sufficiently separated throughout the con-trol execution. Also, consistently with A2, we assumethat the magnitudes of the errors generated by thedelays are bounded by the distances between agentsin such a way that |∑j∈Nt

∑

k∈NtǫTjkcjk| < M/p and

|∑j∈Nt

∑

k∈NtǫTjkcjk

⊥| < M/p for some p > 2.

A4: The magnitude of the acceleration of any givenagent is bounded by a finite valueAmax > 0 for all t > t0,where t0 is the system’s initial time.

We enunciate next a Lemma that will be used in thesubsequent proof of our global stability result.

Lemma 4 Consider the multiagent system evolving ac-cording to the strategy (10), with Gf being a completegraph, and subject to a worst-case point-to-point time-delay D. If A1-A4 hold, the magnitude of the error dueto the delays satisfies:

||e|| ≤ 2(N − 1)Ke

∑

j

||qj(t−D)||(D +Amax ·D2/2),

withKe = 1+(4(N−1)·maxj,k ||cjk||·maxi ||∑

j cji||)/M .

PROOF. Wewill look for a bound on the global magni-tude of the error caused by the delays, considering sepa-rately for each agent the errors affecting its computationof the rotationmatrix and those perturbing its measuredrelative position vectors. As in the previous proof, notethat all the sums that follow are carried out over all theelements in Nt. In addition, we omit the dependence ofvariables on time except when necessary, for clarity rea-sons. From (10) and (20), the input for every agent can

6

d

d

d

Fig. 3. Representation of the error vector eRi due to delays.The variables are expressed in an arbitrary global frame.

be expressed as follows:

qi(t) = Kc ·

∑

j

(qji(t) + ǫji(t))−Rdi (T

di )∑

j

cji

.

(22)

where T di =

∑

j

∑

k(qT

jk(t)+ǫTjk(t))c⊥

jk∑

j

∑

k(qT

jk(t)+ǫT

jk(t))cjk

. Now, we define an

error vector due to the error in the rotation matrix com-puted by agent i as:

eRi = R∑

j

cji −Rdi

∑

j

cji. (23)

R(T ) is the rotation matrix at time t unaffected by de-

lays, i.e. T =

∑

j

∑

kqTjk(t)c

⊥

jk∑

j

∑

kqTjk(t)cjk

. On the other hand, the

errors in the relative position measurements result in thefollowing error vector:

eqi =∑

j

ǫji, (24)

in such a way that, from (22) and (21), the total errorfor agent i is:

ei = eqi + eRi. (25)

We will now bound the magnitude of eRi, in such a waythat the bound is defined in terms of eqi. Figure 3 illus-trates the geometry behind the error vector eRi. Usingtrigonometry, it can be seen that:

||eRi|| = 2| sin(αdi /2)| · ||

∑

j

cji||, (26)

where αdi is the difference between the rotation angles

encapsulated by R (denoted as β) and Rdi (denoted as

βdi ). These two angles are expressed in a common global

frame. Let us choose, without loss of generality, a frame

where R = I2, i.e. β = 0. Then, βdi = αd

i and, consider-ing (7), one can see that:

T di = tanβd

i = tanαdi =

∑

j

∑

k ǫTjkc

⊥jk

∑

j

∑

k(qTjk + ǫTjk)cjk

≤2|∑j

∑

k ǫTjkc

⊥jk|

M, (27)

where the inequality is due to A3, choosing p = 2. Nowobserve that:

|∑

j

∑

k

ǫTjkc⊥jk| ≤ max

j,k||cjk||

∑

j

∑

k

||ǫjk||, (28)

and that we can also write:

∑

j

∑

k

||ǫjk|| ≤∑

j

∑

k

||ǫji||+||ǫki|| = 2(N−1)∑

j

||ǫji||,

(29)where a constant (N − 1) appears instead of N due tothe fact that ||ǫii|| = 0 ∀i. Thus, substituting (28) and(29) in (27), we reach the following bound for T d

i :

T di ≤

4(N − 1)maxj,k ||cjk||∑

j ||ǫji||M

. (30)

Now, we can write:

| sin(αdi /2)| ≤ |αd

i |/2 = | arctan(T di )|/2 ≤ |T d

i |/2, (31)

and therefore, substituting (31) and (30) in (26), we get:

||eRi|| ≤ |T di | · ||

∑

j

cji||

≤4(N − 1)maxj,k ||cjk||maxi ||

∑

j cji||M

∑

j

||ǫji||,

(32)

which holds ∀i. Finally, we obtain the following boundfor the total magnitude of the error vector associatedwith agent i:

||ei(t)|| ≤ ||eqi(t)||+ ||eRi(t)|| ≤ Ke

∑

j

||ǫji(t)||, (33)

withKe = 1+(4(N−1)·maxj,k ||cjk||·maxi ||∑

j cji||)/M .The two terms including a maximum in this expressiondepend on the specific geometry of the desired configu-ration. In particular, they are the maximum interagentdesired distance and the largest sum of the desiredvectors from one agent to all the others. These termsare always greater than zero. We then have that themagnitude of the system’s global error is:

||e|| ≤ Ke

∑

i

∑

j

||ǫji||. (34)

7

From A1, ||ǫji|| is a continuous function. In addition,each relative-position error ǫji emerges from the motionthat j and i have performed in the delay interval (20).The distance traveled by every robot is limited by Amax

by assumption A4, which gives the bound:

∑

i

∑

j

||ǫji(t)|| ≤∑

i

∑

j

∫ t

t−D

||qji(τ)||dτ

≤∑

i

∑

j

∫ t

t−D

(||qj(τ)|| + ||qi(τ)||)dτ

≤ 2(N − 1)∑

j

∫ t

t−D

||qj(τ)||dτ

≤ 2(N − 1)∑

j

||qj(t−D)||(D +Amax ·D2/2), (35)

where a constant (N−1), instead ofN , appears because||qii(τ)|| = 0 ∀i. (34) and (35) result in the statementof the Lemma. 2

Theorem 5 Consider the multiagent system evolvingaccording to (10), with Gf being a complete graph, andsubject to a worst-case point-to-point time-delay D. IfA1-A4 are satisfied and D satisfies the following expres-sion, which is a function of parameters of the system:

D <

(√

1 +Amax

Kc(1 +B)Ke(N − 1)N√N

− 1

)

/Amax,

(36)then the system converges asymptotically to the desiredconfiguration.

PROOF. We will propose a Lyapunov function and,from the expression of its dynamics, state a stabilitycondition that relates the error caused by delays, ||e||,with the non-delayed vector norms ||di||, see (21). Then,using the bound provided in Lemma 4 and exploiting aconstraint inspired by a theorem for stability of time-delay systems, we will obtain an upper bound forD thatensures the stability condition holds.

We define the following candidate Lyapunov function forthe system:

V =1

2N

∑

i

||di||2. (37)

Notice that V is globally positive definite and radiallyunbounded. In addition, V = 0 ⇔ qij = Rcij ∀i, j ∈Nt (i.e. the agents are in the desired configuration). Wecan express the function’s time derivative, consideringseparately the terms depending on T , as follows:

V =∑

i

(

∂V

∂qi

)T

qi +∂V

∂TT =

∑

i

diT qi, (38)

where we have used the fact that ∂V∂T

= 0, which can bereadily verified (see Section 3). Substituting (21) in (38),we have:

V = Kc

∑

i

[

−||di||2 + diTei

]

. (39)

From (39), clearly, V < 0 if and only if:

∑

i

diTei <

∑

i

||di||2. (40)

Using that∑

i diTei ≤

∑

i ||di|| · ||e||, the condition (40)is satisfied if:

∑

i

||di|| · ||e|| <∑

i

||di||2, (41)

and since∑

i ||di||2/(∑

i ||di||)2 ≥ 1/N , we get that the

following is a sufficient condition to ensure V < 0:

||e|| <∑

i ||di||N

. (42)

In what follows, we provide guarantees under which (42)holds true. Due to assumption A2, we have from (21)that at time t−D:

∑

i

||qi(t−D)|| ≤ Kc(∑

i

||di(t−D)||+ ||ei(t−D)||)

≤ Kc(1 +B)∑

i

||di(t−D)||, (43)

and by substituting (43) in the expression stated inLemma 4, we obtain:

||e|| ≤

Kc2(N − 1)(1 +B)Ke(D +AmaxD

2

2)∑

i

||di(t−D)||.

(44)

Let us call V (t) = maxτ∈[t−D,t] V (τ). Observe now that

if V (t) < V (t), the fact that V grows at t will not com-promise the stability of the system, since it does notmake V (t) grow. It is thus only required that V < 0whenever V (t) = V (t) (Gu, Kharitonov, & Chen, 2003).This argument is the main idea behind Razumikhin’stheorem for stability of time-delay systems. The condi-tion V (t) = V (t) would be hard for us to use, since itrequires knowledge of the evolution of the system in theinterval [t−D, t]. Let us introduce a different condition.In particular, it is clear that whenever V (t) = V (t), italso holds that V (t) ≥ V (t −D). Then, we can simplyconsider this latter case so as to prove stability, since thecase V (t) = V (t) is included in it. Therefore, we will use

8

the condition that stability is ensured if V (t) < 0 whenit holds that V (t) ≥ V (t−D) .

To enforce this stability condition, we first use thatif V (t) ≥ V (t − D), then

∑

i ||di(t)|| ≥∑

i ||di(t −D)||/

√N . This can be readily seen from (37). Thus, by

substituting the latter expression in (44), we get:

||e||∑

i ||di||≤ Kc2(N − 1)

√N(1 +B)Ke(D +

AmaxD2

2).

(45)

Then, we enforce (42) in order to guarantee V < 0. From(45), it can be readily seen that (42) is satisfied andtherefore, the system is stable, if the parameters of thesystem satisfy the following condition:

Kc(1 +B)Ke(D +Amax ·D2/2)2(N − 1)N√N < 1.

(46)

Thus, by solving (46) with respect to D, we obtain thatthe system is asymptotically stable if the worst-casetime-delay satisfies:

D <

(√

1 +Amax

Kc(1 +B)Ke(N − 1)N√N

− 1

)

/Amax,

(47)which is the statement of the theorem. 2

Remark 6 The bound (47) is conservative. There arevarious reasons for this: for instance, the delays (and theerrors they generate) are assumed to be maximum and toaffect the system in the worst possible manner, and theLyapunov-based stability conditions used are only suffi-cient. Thus, we are overestimating the effects of delay,and it is clear that the system will be able to accommodateworst-case delays considerably larger than the bound weobtained.

Let us provide a more realistic bound for the worst-casedelay in a typical practical scenario. We focus on the casewhere the delays are small compared to the time requiredfor the system to converge with zero delays, which is,from (17), inversely proportional toKcN .We assume theagents do not increase their speed, or do it negligibly, aftert0. Thus, we can consider Amax = 0. We can also usein practice the less restrictive stability condition ||e|| <∑

i∈Nt||di|| instead of the theoretical one (42). We get an

expression analogous to (46), which results in the bound:

Dr < 1/(CrKc(N − 1)√N), (48)

where Cr = 2(1+B)Ke. This new bound for the delay isstill conservative, but it captures better the relationshipbetween the maximum admissible delay and the time re-quired for the system with zero delays to converge. Fig-ure 4 illustrates the behavior of the bounds D and Dr as

1e−5 1e−4 1e−3 1e−2 1e−1 10

2

4

6

8

10

12

14

Kc

Maxim

um

wors

t−case d

ela

ys (

s)

D

Dr

20 40 60 80 1000

0.5

1

1.5

2

N

Maxim

um

wors

t−case d

ela

ys (

s)

D

Dr

Fig. 4. Maximum delays D, from (47), and Dr, from (48),as a function of Kc -shown in logarithmic scale- (left), andN (right).

−1 0 1

−1.5

−1

−0.5

0

0.5

12

3

4

56

7

8

x (m)

z (

m)

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

5

6

7

8

x (m)

z (

m)

1

24

3

0 200 400 600 800 10000

5

10

15

Time (s)

||v|| (

mm

/s)

and s

cale

d c

ost fu

nction

Fig. 5. Simulation results for the eight-agent circular for-mation. Top: Paths followed by the agents. The final po-sitions are joined by dashed lines. The agents are labeledwith numbers. Bottom-Left: Desired geometric configura-tion. Bottom-Right: Evolution of the norm of the velocityvectors for the eight agents (full lines) and the sum of theircost functions scaled by a constant factor (dashed line).

a function of Kc and N (with the other parameters keptconstant). We see that these bounds decrease as Kc andN grow.

Remark 7 Since ||e|| is generated by the motion of theagents, it is reasonable to assume, as we do in A2, thatthis magnitude will be bounded by that motion, governedby the vectors di. Note that, thanks to allowingB > 1, A2implies that ||e|| can be larger (but not arbitrarily larger)than the sum of norms of di. In essence, A2 states thatwhen all di vanish, ||e|| will vanish too. Observe that thisensures that the agents remain static once they reach thedesired configuration (since V = 0 implies all di = 0,i.e. ||e|| = 0, and in consequence the motion vectors

9

−5 0 5

−4

−2

0

2

4

6

x (m)

z (

m)

−5 0 5

−4

−2

0

2

4

6

x (m)z (

m)

−5 0 5

−4

−2

0

2

4

6

x (m)

z (

m)

50 100 150 200 250 3000

0.05

0.1

0.15

Time (s)

||v|| (

m/s

)

50 100 150 200 250 3000

0.05

0.1

0.15

Time (s)

||v|| (

m/s

)

50 100 150 200 250 300−1.4

−1.3

−1.2

−1.1

−1

−0.9

Time (s)

Ro

tatio

n a

ng

le (

rad

)

Fig. 6. Simulation results for a twenty-agent rectangular grid formation, considering a worst-case time-delay D = 25s. Top:desired configuration (left); agents’ paths with zero delay (center) and with delays (right). The starting positions are markedwith squares, while the final positions are marked with circles. Bottom: time evolution of the magnitudes of the agents’ velocityvectors (thin lines) and their mean value (thick line), with zero delay (left) and with delays (center). Evolution of the formationrotation angles (expressed in a fixed arbitrary global reference frame) estimated by the agents, with delays (right).

perturbed by delays in (21) are null). It is immediate to seethat the final formation is static when global up-to-dateinformation is available (Section 4.1), looking at (10).

Remark 8 Notice that it is ensured that A3 will be sat-isfied if two given agents remain separated (or, equiva-lently, if the agents do not all rendezvous), which is rea-sonable to assume. We analyze next the bounds M and pdefined in this assumption. For small delays, the agentswill reach the desired configuration moving along nearlystraight lines. This steady convergence behavior meansthat typically, M will be equal to the smallest of two val-ues: the initial sum |

∑

j

∑

k qjkT (t0)cjk|, and the value of

the same sum when the desired formation is reached (i.e.when qjk = cjk ∀j, k), which is

∑

j

∑

k ||cjk||2. From as-

sumption A3, given that |∑

j

∑

k qjkT c⊥jk| = 0, we have

that the sum∑

j

∑

k qjkT cjk will be nonzero. Concern-

ing p, it depends on how the errors caused by delays ||ǫjk||compare with the interagent distances ||qjk||. p will belarge for small delays. We can safely choose it as equalto 2 for simplicity.

Remark 9 We can ensure that the acceleration boundAmax (A4) holds by reducing the control gain Kc belowits nominal value, if necessary. Notice that this gain re-duction would imply an increase of the admissible worst-

case delay. Thus, the bound (47) always holds. The min-imum gain achievable will always be positive. To see this,we reason that as the gain tends to zero, the agents wouldstop moving. Therefore, they would be well below the limitAmax, and Kc would not need to decrease further. It isworth highlighting that for any given worst-case delay offinite value, the system can be ensured to be stable bychoosing the control gain Kc appropriately, according to(47) or (48).

5 Simulations

This section presents simulation results to illustrate theperformance of the proposed formation control method.In the first example, we consider a circular formationcomposed of eight agents. As discussed in Section 4, acomplete formation graph is employed. A gain Kc =1e−3 is selected. The results are displayed in Fig. 5. Thepaths followed by the agents from arbitrary initial posi-tions using our control strategy are rectilinear and showexponential convergence to the desired configuration, astheoretically expected. The desired geometric formationis also shown in the same figure. Observe that both thelocation and the orientation of the final agent configura-tion are arbitrary. In addition, we represent the normsof the agents’ velocity vectors, which vanish when con-

10

vergence to the target formation is reached, and the sumof the cost functions computed by all the agents.

The second simulation example illustrates the behaviorof the systemwhen the information used by the agents tocompute their control inputs is affected by time-delays.This case is interesting from a practical perspective,since it corresponds to a scenario where the agents ob-tain directly only local information, and acquire the restof the information through multi-hop communications.It gives rise to a truly distributed framework. The to-tal delay that a message experiences until it reaches itsintended destination is equal to the sum of the delaysincurred as the message travels through the individuallinks in the network on its way to its destination.We con-sider in the simulations that the transmission of informa-tion between every pair of agents has a different total de-lay, and that all these delays are constant and containedin the interval (0, D). We also assume the agents havesynchronized clocks, a common practical requirementin networked multiagent systems (Cortes, Martınez, &Bullo, 2006; Mesbahi & Egerstedt, 2010; Schwager, Ju-lian, Angermann, & Rus, 2011). We consider that ev-ery relative position measurement in the network hasan associated time stamp. These time stamps are usedby each agent to integrate consistently its own measure-ments with those received from its direct neighbors whencomputing its estimates of the relative positions of non-direct neighbors.

Figure 6 depicts the results for an example where the de-sired formation shape is a rectangular grid composed oftwenty agents. We compare the performance of the sys-tem with no delays and with a worst-case delay of D =25s. The same initial configuration is considered in bothcases. The control gain is chosen as Kc = 1e− 3. Whenintroducing the time-delays, we must specify the initialconditions, i.e. the state of the system for t ∈ (−D, 0).We choose the agents to be static during that interval.The effects of the delays on the trajectories can be ob-served in the plots showing the agents’ paths and veloci-ties. In this case, the delays are small and the disturbancethey generate is not very significant. Observe that in thepresence of delays, the agents eventually reach an agree-ment in the rotation angles they compute (expressed ina common frame). The bounds described in section 4.2are: D = 0.01s., obtained from (47) considering B = 2and neglecting Amax, and Dr = 0.2s., computed from(48) using Cr = 60. As stated previously, the system isstable for worst-case delays larger than these bounds.

Figure 7 illustrates how the system’s performance de-grades as the worst-case time-delay increases. A wedge-shaped ten-agent desired formation is used in this case.The gain is Kc = 3e− 3. It can be seen that when D in-creases, the Lyapunov function (37) eventually becomesnon monotonic. In addition, for large delays (larger than50s. in this case) the system is not stable. The perfor-mance of the system remains close to that of the zero-

0 20 40 600

150

500

1e3

1.5e3

Worst−case delay (s)

Tim

e t

o c

on

ve

rge

nce

(s)

0 100 200 3000

5e3

1e4

1.5e4

2e4

2.5e4

Time (s)

V

D=60 sD=45 s

D=0 s

D=15 s

D=30 s

Fig. 7. Simulation results for a ten-agent wedge-shaped for-mation. Left: Time to convergence vs. the worst-case delayD. Right: Evolution of the Lyapunov function V for D equalto 0, 15, 30, 45 and 60s.

delay case as long as D remains small compared to thetime to convergence without delays. The figure also il-lustrates how the degradation of the convergence speedgets worse asD becomes larger. The time to convergenceshown is estimated as the one at which the remainingcontrol error is less than 1% of the initial one. For thisexample, the bounds described in section 4.2, computedas in the previous simulations, have the following values:D = 0.02s. (47) and Dr = 0.19s. (48).

6 Discussion and Conclusion

We have presented a control method through which agroup of mobile agents can be stabilized to a desired ge-ometric configuration. The approach employs the rela-tive interagent positions for this task, and its relevancelies in the fact that it does not require a global com-mon coordinate system or orientation agreement amongthe agents. In this paper, we focused in particular ona scenario in which all the agents have complete infor-mation of the system, obtained through multi-hop com-munications. This is practically feasible for a numberof agents in the hundreds with state-of-the-art technol-ogy in wireless communications and mobile computing.Observe that, even if complete information is employed,the fact that it is expressed locally represents a signif-icant difficulty if a coordinated behavior is desired, asdiscussed in the introduction of the paper. The use offull system information means that our method providesperformance advantages, e.g. in terms of speed of con-vergence, when compared with locally-optimal, partialinformation-based approaches. For the sake of scalabil-ity and flexibility, it would be interesting to make the useof information distributed in our scheme, i.e. to requireeach agent to know the relative positions of only a subsetof the other agents. We have observed in extensive sim-ulations that the presented method can be successfullyextended, in some cases, to scenarios with distributedinformation (i.e. incomplete formation graphs). In par-ticular, the stability of the controller is observed to de-pend on both the characteristics of the formation graphand the geometry of the desired configuration. In light of(10), stabilization to the desired formation in this casemay be posed as a nonlinear consensus problem where

11

agreement on the rotation matrices Ri is required.

Our future work will consider the case in which theagents use only partial information of the group, corre-sponding to a subset of neighbors. The final objective isto characterize the classes of formation graphs and de-sired geometric configurations for which the proposedmethod can be effective. In addition, another relevantissue to explore is the consideration of nonholonomickinematic agents.

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